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Let $R$ be a ring and let $I$ be a proper ideal of $R$.

If $R/I$ has no zero divisors, then is it true that $R$ has no zero divisors?

My attempt: Suppose $R$ has zero divisors, say $ab=0$ for some $a,b\in R^*$. Then $(a+I)(b+I)=ab+I=I$. However, I cannot exclude the case where $a,b\in I$.

Any ideas?

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Hint: every nontrivial ring has a maximal ideal –  Marcin Łoś Feb 23 at 8:41
    
So what when $I$ is not the maximal ideal? still confused... –  user130893 Feb 23 at 9:50
    
When $I$ is a maximal ideal, what do we know about the quotient? –  Marcin Łoś Feb 23 at 9:51
    
Then the quotient is a simple ring. –  user130893 Feb 23 at 9:55
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It is also a field, hence it has no zero divisors. We can thus take arbitrary ring with zero divisors, and create a quotient which has none. –  Marcin Łoś Feb 23 at 9:58

1 Answer 1

If you try to prove something and hit a wall, then you should switch to finding a counterexample for a while. Then if that doesn't work, try proving again. Back and forth you go.

Really you should find a counterexample right away. You could, for example, take a look at $\Bbb Z/\Bbb Z4$.

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